Difference between revisions of "Semigroup property of delta exponential"

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<strong>[[Semigroup property of delta exponential|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], $t,s \in \mathbb{T}$, and let $p \in \mathcal{R}\left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:
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<strong>[[Semigroup property of delta exponential|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], let $t,s \in \mathbb{T}$, and let $p \in \mathcal{R}\left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:
 
$$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$
 
$$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$
 
where $e_p$ denotes the [[delta exponential]].
 
where $e_p$ denotes the [[delta exponential]].

Revision as of 23:14, 31 May 2016

Theorem: Let $\mathbb{T}$ be a time scale, let $t,s \in \mathbb{T}$, and let $p \in \mathcal{R}\left( \mathbb{T},\mathbb{C} \right)$ be a regressive function. The following formula holds: $$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$ where $e_p$ denotes the delta exponential.

Proof: